#149187

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1850
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,4)(5,6,10,17,11,15,31,41,12,16,32,42,9,28,30,55)(7,13,20,36,19,37,46,62,21,35,47,61,29,34,56,60)(8,22,26,49,27,40,54,63,25,50,53,66,24,14,38,39)(18,43,45,64,44,33,57,58,23,48,52,65,51,59,67,68);(1,2,3,4)(5,7)(6,13)(8,23)(9,29)(10,20)(11,19)(12,21)(14,33)(15,37)(16,35)(17,36)(18,25)(22,48)(24,44)(26,52)(27,51)(28,34)(30,56)(31,46)(32,47)(38,57)(39,58)(40,59)(41,62)(42,61)(43,50)(45,53)(49,65)(54,67)(55,60)(63,68)(64,66);(5,8,9,24,12,25,11,27)(6,14,15,22,16,40,28,50)(7,18,19,44,21,23,29,51)(10,26,30,38,32,53,31,54)(13,33,34,43,35,59,37,48)(17,39,41,49,42,63,55,66)(20,45,46,57,47,52,56,67)(36,58,60,64,61,68,62,65)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56383);